cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

A365559 Number of free n-polysticks (or polyedges) in 3 dimensions.

Original entry on oeis.org

1, 2, 7, 28, 160, 1085, 8403, 69824, 607988, 5448444, 49846437, 462977928
Offset: 1

Views

Author

Pontus von Brömssen, Sep 09 2023

Keywords

Comments

a(1)-a(8) verified and a(9)-a(10) computed by John Mason.

Examples

			There are a(3) = 7 free 3-polysticks in 3 dimensions: A019988(3) = 5 properly 1- or 2-dimensional (straight, "U", "T", "L", and skew, similar to the 5 tetrominoes) and 2 properly 3-dimensional (one path-like and one with a vertex of degree 3).

Links

Crossrefs

Sum of first three columns of A365566.
Cf. A019988 (2 dimensions), A365560 (fixed), A365561 (4 dimensions), A365563 (5 dimensions), A365565 (arbitrary dimension).
14th row of A366766.

Extensions

a(11) derived from Ishino Keiichiro's website (sum of 2-sided 2D-edges and 3D-edges), added by Pontus von Brömssen, Dec 21 2023
a(12) from John Mason, Mar 07 2025

A365561 Number of free n-polysticks (or polyedges) in 4 dimensions.

Original entry on oeis.org

1, 2, 7, 31, 199, 1651, 16648
Offset: 1

Views

Author

Pontus von Brömssen, Sep 09 2023

Keywords

Links

Crossrefs

42nd row of A366766.
Sum of first four columns of A365566.
Cf. A019988 (2 dimensions), A365559 (3 dimensions), A365562 (fixed), A365563 (5 dimensions), A365565 (arbitrary dimension).

A365563 Number of free n-polysticks (or polyedges) in 5 dimensions.

Original entry on oeis.org

1, 2, 7, 31, 205, 1768
Offset: 1

Views

Author

Pontus von Brömssen, Sep 09 2023

Keywords

Links

Crossrefs

Sum of first five columns of A365566.
158th row of A366766.
Cf. A019988 (2 dimensions), A365559 (3 dimensions), A365561 (4 dimensions), A365564 (fixed), A365565 (arbitrary dimension).

A365565 Number of free n-polysticks (or polyedges) in arbitrary dimension.

Original entry on oeis.org

1, 2, 7, 31, 205, 1779
Offset: 1

Views

Author

Pontus von Brömssen, Sep 09 2023

Keywords

Links

Crossrefs

Row sums of A365566.
Cf. A005519, A019988 (2 dimensions), A365559 (3 dimensions), A365561 (4 dimensions), A365563 (5 dimensions).

A385582 Triangle read by rows: T(n,d) is the number of fixed, properly d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 4, 1, 20, 32, 1, 86, 420, 400, 1, 370, 4164, 10368, 6912, 1, 1626, 38205, 186552, 301840, 153664, 1, 7310, 343380, 2934560, 8637760, 10223616, 4194304, 1, 33464, 3086049, 43517697, 207353960, 427708848, 396809280, 136048896
Offset: 1

Views

Author

Pontus von Brömssen, Jul 04 2025

Keywords

Examples

			Triangle begins:
  n\d| 1     2       3        4         5         6         7         8
  ---+-----------------------------------------------------------------
  1  | 1
  2  | 1     4
  3  | 1    20      32
  4  | 1    86     420      400
  5  | 1   370    4164    10368      6912
  6  | 1  1626   38205   186552    301840    153664
  7  | 1  7310  343380  2934560   8637760  10223616   4194304
  8  | 1 33464 3086049 43517697 207353960 427708848 396809280 136048896

Links

Crossrefs

Cf. A127670 (main diagonal), A195739 (polyominoes), A365566 (free), A385581.

Formula

T(n,d) = Sum_{k=1..d} (-1)^(d-k)*binomial(d,k)*A385581(n,k).

A385583 Triangle read by rows: T(n,d) is the number of free d-dimensional polysticks of size n.

Original entry on oeis.org

1, 1, 2, 1, 5, 7, 1, 16, 28, 31, 1, 55, 160, 199, 205, 1, 222, 1085, 1651, 1768, 1779
Offset: 1

Views

Author

Pontus von Brömssen, Jul 04 2025

Keywords

Comments

If d > n, there are T(n,n) such polysticks. The triangle only includes the values for d <= n.

Examples

			Triangle begins:
  n\d| 1   2    3    4    5    6
  ---+--------------------------
  1  | 1
  2  | 1   2
  3  | 1   5    7
  4  | 1  16   28   31
  5  | 1  55  160  199  205
  6  | 1 222 1085 1651 1768 1779

Links

Crossrefs

Cf. A330891 (polyominoes), A365565 (main diagonal), A365566, A385581 (fixed).
Columns: A019988 (d=2), A365559 (d=3), A365561 (d=4), A365563 (d=5).

Formula

T(n,d) = Sum_{k=1..d} A365566(n,k).

A387004 Triangle read by rows: T(n,d) is the number of free, properly d-dimensional (d,2)-polyominoids of size n, 2 <= d <= n+1.

Original entry on oeis.org

1, 1, 1, 2, 7, 3, 5, 49, 41, 8
Offset: 1

Views

Author

Pontus von Brömssen, Aug 14 2025

Keywords

Examples

			Triangle begins:
  n\d| 2  3  4  5
  ---+-----------
  1  | 1
  2  | 1  1
  3  | 2  7  3
  4  | 5 49 41  8

Links

Crossrefs

Cf. A000105 (column d=2), A049430 (polyominoes), A365566 (polysticks), A387002 (fixed), A387003, A387005 (row sums).

Formula

T(n,d) = A387003(n,d) - A387003(n,d-1) (with A387003(n,1) = 0).
Showing 1-7 of 7 results.

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